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Question
Evaluate the following limit:
`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`
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Solution
`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`
= `lim_(y -> -3) (((y^5 + 243)/(y + 3)))/(((y^3 + 27)/(y + 3))) ...[(because y -> -3"," therefore y ≠ -3","),(therefore y + 3 ≠ 0)]`
= `(lim_(y -> -3)[(y^5 - (- 3)^5)/(y - ( - 3))])/(lim_(y -> - 3)[(y^3 - ( - 3)^3)/(y - (- 3))]`
= `(5(-3)^4)/(3(-3)^2) ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "n"*"a"^("n" - 1)]`
= `5/3 xx 9`
= 15
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